Question

In: Advanced Math

1) Show that if A is an open set in R and k ∈ R \...

1) Show that if A is an open set in R and k ∈ R \ {0}, then the set kA = {ka | a ∈ A} is open.

Expert Solution

1. Given is an open set and

To show that open we need to show that every point of is an interior point of .

Let be an arbitrary point .

As is open and so there exist such that

Now we will show that

let

then as .

So

If we choose the .

So for each there exist such that that is is an interior point of . As is arbitrary so each point of is an interior point of .

Hence is open .

.

If you have any doubt or need more clarification at any step please comment.

Colby Messinger answered 3 years ago

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